The Germans would probably be butchered in the event of an invasion. It would probably result in another situation like Greece--too many soldiers concentrated on Switzerland would drag into troop reserves meant for the Soviet Union, which would mean that the Soviets have a just slightly easier time. Heavy Swiss defenses, coupled with brutal mountainous terrain, would make things awful on the Germans.
Just because not ALL of Switzerland is in heavy mountain fortresses, doesn't mean that the rest of the country is an open door. Much of the less formidable regions of the country have restricted access from invaders, and the Swiss Army has a fair number of excellent infantry divisions. The whole army did not consist of relatively immobile fortress troops.
AISI, the greatest danger for the Swiss is if they fall prey to the same thinking that befell the Norwegians, the Dutch, the Belgians, and even the Soviets: Just play it meek enough, drive your head in the sand long enough, and somehow the Nazi threat will just all go away. So the Nazis invade Switzerland, and between sabotage by would-be collaborators (remember what happened to Norway) and having the army poorly mobilized and out of their prepared defensive positions, the Nazis could get more of the country at the very start than anyone would have expected. Add possible demoralization to this, and a mostly quick occupation is not ASB.
If Switzerland manages to hold off the German invasion, there's at least a halfway decent chance that they'll join the Allies.
I don't want to be sounding snarky

, but with the exception of the Soviets (ruled by the hyper-paranoid Stalin, anytime that a neutral nation is invaded by one side in a general war the invaded country instantly becomes a full military ally of the invader's active belligerent enemies. That's just political reality.
Otherwise, the Swiss people themselves will be demanding their government join the Allies. That frees up Swiss air bases for shuttle bombing to German targets by the Western Allies and even using fighter bases. The Allies won't get immediate air superiority, but depending on when this happens ITTL, it could represent a serious threat to German airspace years before long-range Allied fighters become available.
Not to mention the Allies can now flank the Rhine River when that becomes an issue.
Whether that leads to the Swiss joining NATO afterwards I can't say--Switzerland may no longer see a threat to itself, but other former neutrals like Portugal, Norway, and Denmark joined, which at least leaves the door open.
I'd say that the better the Nazis do against the Swiss, and the worse the Nazis do unto them, the more likely Switzerland joins NATO. A Nazi conquest of Switzerland makes for the lesson that you can't go home again.
Also, the use of Switzerland as a base for Allied soldiers would probably make the invasion of Germany a bit easier. Even if Switzerland fell, you would probably see a pretty large and active resistance movement.
IMHO, if Switzerland fell outright, you'd probably see the country bypassed.
More realistically, everything worth overruning is overran by Germans in three to seven days<snip>
As I posted above, the only way the Swiss fall that quickly is due to the Germans pulling off a "Norway" in terms of Swiss confusion.
Is it just me, or do you have something against the Swiss?
Like with Americans, people tend to have a problem with Swiss politics, not the Swiss people themselves. In the case of Switzerland's politics (and laws), its their history as misery merchants. Blood diamonds, banking with no questions, accepting Nazi loot, neutrality uber alles...
Yeah, but the cannibalism part is rather wacky.
The Swiss troops would surrender before the cannabilism started. They are not isolated fanatical Japanese, and even for them that was a rarity.